A vice IS a press, it just uses an inclined plane (the screw attached to the handle) to move one jaw against the other instead of hydraulics like in most presses.
When pressing bearings in or out, to prevent damage to the bearing you must always put force on the race that has the resistance. In other words, if it's the outer race (race is the grooved part of the bearing, always in two's, an inner and an outer on circular bearings like you are talking about) that is going to hold the bearing in the pulley, you put pressure on that outer race. If the inner race will do the holding, say on a shaft, you would put pressure on the inner race.
Try to avoid putting pressure "through" a bearing, such as on the inner race to seat a bearing that is held in by the outer race. Bearings aren't designed to withstand that kind of axial load.
In the pulley that will get the bearing, try and support the pulley as close to the bearing as possible. The further the support is from the bearing, the easier the pulley will deform.
Have I described it well enough or is it as clear as mud now?
Tom